How can I customize the Render Button in the Render GUI to execute a custom script
The easiest way is to right click on the render button (or any UI item) and select Edit Source. You will see a message in the info header to 'See properties_render.py in the text editor'. Simply change one of your windows to a text editor and select the properties_render.py from the text block list. You will find the cursor located on the line used for the UI item you selected.

If you search the blender python api documentation for UILayout you will find the operator function used there takes the operator name as the first argument. So you would change "render.render" to the name of the operator you want to use.

Remember to save the edited text block back to the text file. Also this file will be replaced with each blender upgrade so creating an addon may be a wiser choice for something you plan to keep using.